We finish this section by explaining why we believe that
Proposition REF  is the conceptual reason
for the results in Section REF  stating that the associated module functors
for the classical and deformed Hopf fibration are naturally isomorphic.
The following argument is inspired by [1]}.
Let us first note that the underlying vector space of the Hopf algebra \(H = \mathcal {O}(U(1))\)  admits a decomposition
\(H = \bigoplus _{n\in \mathbb {Z}} C_n\) , where \(C_n =\mathbb {C}\,t^n\)  is the
(1-dimensional) vector space spanned by the \(n\) -th power of the generator \(t\) .
(Recall that \(t^{-1} = t^\ast \) .)
As \(\Delta (t^n)= t^n\otimes t^n\) ,
each \(C_n\)  is a left \(H\) -comodule via the coaction \(\Delta \colon C_n\rightarrow H\otimes C_n\) .
It follows that \(A \cong \bigoplus _{n\in \mathbb {Z}} A\square _H C_n\) 
is a direct sum of associated modules \(E_A(C_n) = A\square _H C_n\) 
with \(C_n\in {}^H\!\!{M}{}\!_{\mathrm {fin}} \)  finite-dimensional. By [2]},
each \(E_A(C_n) \)  is a finitely generated projective left \(B\otimes \overline{\mathcal {O}}(\mathbb {R})\) -module
and hence it defines an element \([E_A(C_n)]\)  of the
zeroth \(K\) -theory group \(K_0\big (B\otimes \overline{\mathcal {O}}(\mathbb {R})\big )\) . Analogously, there exists a decomposition
\(A_p \cong \bigoplus _{n\in \mathbb {Z}} A_p \square _H C_n\)  into a direct sum
of associated modules \(E_{A_p}(C_n) = A_p\square _H C_n\) 
and we obtain elements \([E_{A_p}(C_n)]\in K_0(B)\) , for \(p=0,1\) .
Because of Proposition REF ,
we know that \([E_{A_p}(C_n)]\in K_0(B)\)  is the image under
\(K_0({\mathrm {ev}_p}_\ast ) \colon \ K_0\big (B \otimes \overline{\mathcal {O}}(\mathbb {R})\big ) \longrightarrow K_0(B)\) 
